All Questions
286 questions
39
votes
2
answers
5k
views
Why is there a connection between enumerative geometry and nonlinear waves?
Recently I encountered in a class the fact that there is a generating function of Gromov–Witten invariants that satisfies the Korteweg–de Vries hierarchy. Let me state the fact more precisely. ...
- 1,589
18
votes
0
answers
987
views
"Special" meanders
One of the open problems in combinatorics is enumeration of meanders.
Here on MO I only could find them under the heading not-especially-famous-long-open-problems-which-anyone-can-understand.
Since my ...
- 18.4k
79
votes
12
answers
13k
views
Is there a high-concept explanation for why characteristic 2 is special?
The structure of the multiplicative groups of $\mathbb{Z}/p\mathbb{Z}$ or of $\mathbb{Z}_p$ is the same for odd primes, but not for $2.$ Quadratic reciprocity has a uniform statement for odd primes, ...
- 118k
21
votes
3
answers
2k
views
Invariants for the exceptional complex simple Lie algebra $F_4$
This is an edited version of the original question taking into account the comments below by Bruce. The original formulation was imprecise.
Let $\mathfrak{g}$ denote a complex simple Lie algebra of ...
- 33.3k
12
votes
1
answer
980
views
How are Sheffer polynomials related to Lie theory?
Sheffer polynomials $\{P_n(x)\}$ have generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra correspondence. Is there any ...
28
votes
3
answers
6k
views
Motivation of Virasoro algebra
I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with ...
- 1,663
7
votes
1
answer
1k
views
When do two non-degenerate quadratic forms give rise to isomorphic Lie algebras?
Let $V$ be a vector space over some number field $k$. (I'm fine with $\mathbb{Q}$.)
Let $\phi \colon V \to k$ be a non-degenerate quadratic form. Associated with $\phi$ is the orthogonal group $\...
- 5,504
17
votes
1
answer
502
views
Irreducibility of root-height generating polynomial
The height $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum_{i=1}^n c_i$ where $\alpha = \sum_{i=1}^n c_i \alpha_i$ is its decomposition as a sum of ...
- 2,753
5
votes
2
answers
505
views
A finiteness property for semi-simple algebraic groups
Let $G$ be a semi-simple algebraic group over a field $K$, I am considering a question about whether there exists a finite set of semi-simple $K$-subgroups, say $H_1,...,H_r$, such that for any semi-...
- 199
3
votes
3
answers
637
views
Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation
The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry,
$...
- 10.5k
77
votes
7
answers
21k
views
What is the symbol of a differential operator?
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
- 54.6k
70
votes
11
answers
13k
views
What is significant about the half-sum of positive roots?
I apologize for the somewhat vague question: there may be multiple answers but I think this is phrased in such a way that precise answers are possible.
Let $\mathfrak{g}$ be a semisimple Lie algebra (...
- 3,605
53
votes
5
answers
8k
views
Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
- 1,422
34
votes
2
answers
3k
views
The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.
I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...
- 41.4k
22
votes
1
answer
2k
views
Modern reference for maximal connected subgroups of compact Lie groups
What's the nicest place to see a list of the maximal connected subgroups of compact Lie groups? Is there anything on-line?
I looked at Tits' Bourbaki talk on Dynkin's and others' work, but he admits ...
- 27.9k
19
votes
2
answers
2k
views
Dual versions of "folding" symmetric ADE Dynkin diagrams?
Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams ...
- 52.9k
18
votes
3
answers
3k
views
What is a homotopy between $L_\infty$-algebra morphisms
A $L_\infty$-algebra can be defined in many different ways. One common way, that
gives the 'right' kind of morphisms, is that a $L_\infty$-algebra is a graded cocommutative and coassociative ...
- 2,074
16
votes
3
answers
4k
views
Generators of invariant polynomials of semisimple Lie algebra
Suppose $\mathfrak{g}$ is a complex semi-simple Lie algebra. By a theorem of Chevalley, we know that $S(\mathfrak{g})^\mathfrak{g}$, i.e. the $\mathfrak{g}$ invariant polynomials, is generated by $l$ ...
- 587
12
votes
4
answers
866
views
Breaking up the free Lie algebra into GL irreps
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\L{\mathfrak{L}}$The free Lie algebra $\L(V)$ generated by an $r$-dimensional vector space $V$ is, in the
language of https://en.wikipedia.org/wiki/...
- 8,336
10
votes
2
answers
2k
views
Chevalley Groups over an arbitrary ring.
My question is simply about the Chevalley groups over rings. In many books, including Carter's book on "Simple groups of Lie types", the groups are considered over fields. I have checked the ...
- 2,508
10
votes
4
answers
2k
views
Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$
One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...
- 3,541
106
votes
3
answers
10k
views
Has the Lie group E8 really been detected experimentally?
A few months ago there were several math talks about how the Lie group E8 had been detected in some physics experiment. I recently looked up the original paper where this was announced,
"Quantum ...
- 20.7k
60
votes
8
answers
13k
views
Why the Killing form?
I'm teaching a short summer course on algebraic groups and it's time to talk about the Killing form on the Lie algebra. The students are all undergrads of varying levels of inexperience, and I try to ...
- 7,273
58
votes
3
answers
4k
views
Is "semisimple" a dense condition among Lie algebras?
The "Motivation" section is a cute story, and may be skipped; the "Definitions" section establishes notation and background results; my question is in "My Question", and in brief in the title. Some ...
- 54.6k
54
votes
5
answers
10k
views
Motivating the Casimir element
Weyl's theorem states that any finite-dimensional representation of a finite-dimensional semisimple Lie algebra is completely reducible. In my mind, the "natural" way to prove this result is by way ...
- 82.7k
45
votes
16
answers
8k
views
What does the generating function $x/(1 - e^{-x})$ count?
Let $x$ be a formal (or small, since the function is analytic) variable, and consider the power series
$$ A(x) = \frac{x}{1 - e^{-x}} = \sum_{m=0}^\infty \left( -\sum_{n=1}^\infty \frac{(-x)^n}{(n+1)!}...
- 54.6k
42
votes
9
answers
6k
views
Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?
Harold Williams, Pablo Solis, and I were chatting and the following question came up.
In Lie group land (where you're doing differential geometry), given a finite-dimensional Lie algebra g, you can ...
- 24.1k
38
votes
18
answers
24k
views
Learning about Lie groups
Can someone suggest a good book for teaching myself about Lie groups? I study algebraic geometry and commutative algebra, and I like lots of examples. Thanks.
33
votes
3
answers
6k
views
When is a finite dimensional real or complex Lie Group not a matrix group
I have a smattering of knowledge and disconnected facts about this question, so I would like to clarify the following discussion, and I also seek references and citations supporting this knowledge. ...
- 1,161
30
votes
4
answers
11k
views
Outer automorphisms of simple Lie Algebras
There is, of course, a complete classification for simple complex Lie algebras. Is there a good reference which lists the group of outer automorphisms for each?
- 1,233
30
votes
4
answers
3k
views
A mysterious Heisenberg algebra identity from Sylvester, 1867
I am trying to understand two papers by James Joseph Sylvester:
P92: "Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, analogues, and laws of ...
- 33.8k
28
votes
1
answer
1k
views
What's the status of the following relationship between Ramanujan's $\tau$ function and the simple Lie algebras?
Qiaochu asked this in the comments to this question. Since this is really his question, not mine, I will make this one Community Wiki. In MR0522147, Dyson mentions the generating function $\tau(n)$ ...
26
votes
1
answer
2k
views
Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?
The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...
- 28.1k
22
votes
6
answers
3k
views
Automorphism group of real orthogonal Lie groups
I would like to understand what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$ and $pq \neq 0$. My working definition of $Out$ is as follows:
Let us denote by $...
- 2,818
19
votes
5
answers
2k
views
Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: $...
- 43.2k
19
votes
3
answers
2k
views
Elkies' supersingularity theorem in higher dimension
The following is a theorem of Elkies:
Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero.
...
- 156k
18
votes
3
answers
1k
views
Is a retract of a free object free?
I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?
- 1,875
18
votes
2
answers
1k
views
Is homology finitely generated as an algebra?
If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; ...
- 8,727
18
votes
3
answers
1k
views
What happened to the fourth paper in the series "On the classification of primitive ideals for complex classical Lie algebras" by Garfinkle?
In a series of papers in Compositio Math. entitled On the classification of primitive ideals for complex classical Lie algebras I, II and III, Garfinkle describes an algorithm that allows one to ...
- 2,508
18
votes
3
answers
574
views
Does there exist something like an $H_3$ and $H_4$ (icosahedral) Lie algebra or algebraic group?
The (finite-dimensional) complex simple Lie algebras have been classified by Killing and Cartan a long time ago in the $\mathsf{A}_n,\mathsf{B}_n,\mathsf{C}_n,\mathsf{D}_n$ families and $\mathsf{G}_2,\...
- 32.5k
17
votes
1
answer
2k
views
So, did Poincaré prove PBW or not?
This seems to be a question whose answer depends on whom you ask. Maybe we can come up with a final answer?
It is known that Poincaré, at least, invented something that can be called Poincaré-...
- 33.8k
15
votes
3
answers
3k
views
Which is the correct universal enveloping algebra in positive characteristic?
This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers.
Let $\...
- 54.6k
14
votes
1
answer
1k
views
Uniform proof of dimension formula for minimal special nilpotent orbit?
Given a simple Lie algebra over an algebraically closed field of good characteristic such
as $\mathbb{C}$, its subvariety $\mathcal{N}$ of nilpotent elements has dimension $2N$ (where $N$ is the ...
- 52.9k
14
votes
3
answers
1k
views
About enveloping algebras of direct sums
This question is imported from MSE. It is linked to this one in the case of semi-direct products.
My question Let us consider a Lie $R$-algebra ($R$ is a commutative ring) written as a (module) ...
- 3,738
14
votes
1
answer
544
views
Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?
This might be forced to migrate to math.SE but let me still risk it.
The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism ...
- 18.4k
13
votes
3
answers
3k
views
How to Compute the coordinate ring of flag variety?
Let $G$ be algebraic group over $\mathbb(C)$(semisimple), $B$ be Borel subgroup. consider flag variety $G/B$. It is known that $G/B$ is projective variety. So one consider the homogeneous coordinate ...
- 5,515
13
votes
2
answers
1k
views
Can one define quantized universal enveloping algebras in a basis-free way?
(For the background, I am learning about quantum groups — essentially in order to understand crystal/global/canonical bases in the context of this question — from the books by Jantzen and by Hong&...
- 32.5k
11
votes
2
answers
7k
views
How to find Casimir operators?
Given a general Lie algebra, is there a general procedure to find all its Casimir operator?
- 111
10
votes
3
answers
1k
views
subgroup of SU(N) with maximal manifold dimension
Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not ...
- 1,207
9
votes
1
answer
3k
views
Lie algebra semisimple if and only if perfect?
If $L$ is a semisimple lie algebra then $L=[L,L]$. Is the opposite true?
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